Multi-user mixed multi-hop relay network

ABSTRACT

The present disclosure relates to a triple-hop multiuser relay network. Further, the relay network is comprised of mixed communication mediums (radiofrequency/free-space optical/radiofrequency), and utilizes a generalized order user scheduling scheme for determining the next source or destination to be selected for transmission. Closed-form expressions were achieved to describe outage probability, average symbol error probability, and channel capacity assuming Rayleigh and Gamma-Gamma fading models for the radiofrequency and free-space optical links, respectively. The effects of pointing errors on the free-space optical link were also considered. Additionally, a power allocation algorithm was proposed to optimize power allocation at each hop.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims priority to U.S. Provisional ApplicationNo. 62/578,090, filed Oct. 27, 2017, the teaching of which is herebyincorporated by reference in its entirety for all purposes.

STATEMENT REGARDING PRIOR DISCLOSURE BY THE INVENTORS

Aspects of this technology are described in an article “A new scenarioof triple-hop mixed RF/FSO/RF relay network with generalized order userscheduling and power allocation” published in EURASIP Journal onWireless Communications and Networking, on Oct. 28, 2016, which isincorporated herein by reference in its entirety.

BACKGROUND Field of the Disclosure

The present disclosure relates to a multi-hop relay network withgeneralized order user scheduling and transmission power allocation.

Description of the Related Art

Cooperative relay networks present an efficient solution for multipathfading issues in wireless communications. In these relay networks, arelay node or a set of relay nodes facilitate propagation of a messagefrom one or more source nodes to one or more destination nodes, therebyproviding diversity, widening coverage area, and reducing the need forhigh power transmitters. Pursuant to network architecture and desiredfunctionality, each node may employ either an amplify-and-forward (AF)or decode-and-forward (DF) scheme. The simplicity of AF schemes is oftenweighed against the computationally demanding, but improved output, ofDF schemes.

Recent efforts to reduce power consumption, expand coverage and improvereliability of wireless communications have employed a mixture of datatransmission modalities. By employing relays and varied transmissionmodalities, networks are able to increase communication distance andimprove network diversity. To this end, several approaches have beenexplored, including single-relay free space optical (FSO) communicationsand dual-hop mixed radiofrequency (RF) and FSO relay networks. Further,by considering multiple users a network aims to achieve multiuserdiversity.

While recent work has considered triple-hop relaying for only one typeof transmission modality, broad advances have focused on dual-hop mixedRF/FSO relay networks, representative of applications where multipleusers communicate with a relay node via RF links and the relay forwardstheir messages to a base station over an FSO link.

Therefore, one objective of the present disclosure is to provide andevaluate a model of triple-hop mixed-mode relaying in wireless networks.

The foregoing “Background” description is for the purpose of generallypresenting the context of the disclosure. Work of the inventors, to theextent it is described in this background section, as well as aspects ofthe description which may not otherwise qualify as prior art at the timeof tiling, are neither expressly or impliedly admitted as prior artagainst the present invention.

SUMMARY

The present disclosure relates to a triple-hop mixed RF/FSO/RF relaynetwork with generalized order user scheduling and transmission powerallocation.

The triple-hop mixed RF/FSO/RF relay network includes K₁ sources, two DFrelays, and K₂ destinations. The sources and destinations are connectedto respective relay nodes via RF links, while the relay nodes areconnected via FSO link.

The generalized order user scheduling scheme selects the source with theN₁ ^(th) best signal-to-noise ratio (SNR) among the available sources tocommunicate with the first relay node. Similarly, following transmissionvia FSO from the first relay node to the second relay node, thedestination with the N₂ ^(th) best SNR is selected to receive itsmessage from the second relay.

Further, optimum transmission powers of the selected user are obtainedon the first hop, first relay, and second relay.

The foregoing paragraphs have been provided by way of generalintroduction, and are not intended to limit the scope of the followingclaims. The described embodiments, together with further advantages,will be best understood by reference to the following detaileddescription taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the disclosure and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 is a schematic of a multiuser multi-hop mixed transmissionmodality network with generalized order user scheduling according to oneor more aspects of the disclosed subject matter;

FIG. 2 is a schematic of a multiuser triple-hop mixed RF/FSO/RF relaynetwork with generalized order user scheduling according to one or moreaspects of the disclosed subject matter;

FIG. 3 is a graphical representation of system outage probability versusSNR of a multiuser mixed RF/FSO/RF relay network with generalized orderuser scheduling for different values of N₁=N₂ according to one or moreaspects of the disclosed subject matter;

FIG. 4 is a graphical representation of system outage probability versusSNR of a multiuser mixed RF/FSO/RF relay network with generalized orderuser scheduling for different values of K₁=K₂ according to one or moreaspects of the disclosed subject matter;

FIG. 5 is a graphical representation of system outage probability versusSNR of a multiuser mixed RF/FSO/RF relay network with generalized orderuser scheduling for different values of γ_(out) with fixed average SNRsand varying average SNRs according to one or more aspects of thedisclosed subject matter;

FIG. 6 is a graphical representation of system outage probability versusorder of selected user of multiuser mixed RF/FSO/RF relay network withgeneralized order user scheduling for different values of average SNRaccording to one or more aspects of the disclosed subject matter;

FIG. 7 is a graphical representation of system outage probability versusSNR of multiuser mixed RF/FSO/RF relay network with generalized orderuser scheduling for different values of γ_(out) with and without poweroptimization according to one or more aspects of the disclosed subjectmatter;

FIG. 8 is a graphical representation of system average symbol errorprobability versus SNR of a multiuser mixed RF/FSO/RF relay network withgeneralized order user scheduling for different values of ζ according toone or more aspects of the disclosed subject matter;

FIG. 9 is a graphical representation of system average symbol errorprobability versus SNR of a multiuser mixed RF/FSO/RF relay network withgeneralized order user scheduling for different values of α, β, and raccording to one or more aspects of the disclosed subject matter;

FIG. 10 is a graphical representation of system average symbol errorprobability versus SNR of a multiuser mixed RF/FSO/RF relay network withgeneralized order user scheduling for different values of (K₁, N₁) and(K₂, N₂) according to one or more aspects of the disclosed subjectmatter;

FIG. 11 is a graphical representation of ergodic capacity versus SNR ofa multiuser mixed RF/FSO/RF relay network with generalized order userscheduling for different values of N₁=N₂ according to one or moreaspects of the disclosed subject matter; and

FIG. 12 is a hardware block diagram of a server according to one or moreexemplary aspects of the disclosed subject matter.

DETAILED DESCRIPTION

The terms “a” or “an”, as used herein, are defined as one or more thanone. The term “plurality”, as used herein, is defined as two or morethan two. The term “another”, as used herein, is defined as at least asecond or more. The terms “including” and/or “having”, as used herein,are defined as comprising (i.e., open language). Reference throughoutthis document to “one embodiment”, “certain embodiments”, “anembodiment”, “an implementation”, “an example” or similar terms meansthat a particular feature, structure, or characteristic described inconnection with the embodiment is included in at least one embodiment ofthe present disclosure. Thus, the appearances of such phrases or invarious places throughout this specification are not necessarily allreferring to the same embodiment. Furthermore, the particular features,structures, or characteristics may be combined in any suitable manner inone or more embodiments without limitation.

Recently, incorporation of free-space optical communication in relaynetworks has been proposed to provide improved reliability in the‘last-mile’ of wireless communications. These networks, often known asdual-hop relay networks, transmit a source message from a source to arelay node over a RF link (licensed frequencies) and then forward themessage to the destination over an FSO link (license-free). In suchnetworks, relays provide greater diversity among nodes, expand thecoverage area, and reduce the need for high-power transmitters. Further,this approach can be complemented via multiuser cooperation andopportunistic scheduling.

Extensive work has been dedicated to the above, filling the connectivitygap in ‘last-mile’ connectivity while conserving economic resources andsaving bandwidth by exploiting optical communications. To this point,however, FSO communications have been incorporated into dual-hopnetworks with the FSO link connecting directly to a base station at aterminus. Therefore, a triple-hop mixed relay network with generalizeduser scheduling and power allocation algorithm, yet to be developed, isdescribed in the present disclosure.

FIG. 1 shows a triple-hop mixed RF/FSO/RF relay network consisting of K₁sources on a first hop 117 U_(k) (k=1, 2, . . . , K₁) (102, 103, 104),two un-coded type DF relays R_(i) (i=1, 2) (105, 115) and K₂destinations on a third hop 119 D_(j) (j=1, 2, . . . , K₂) (112, 113,114). The sources 110 are connected with a first relay 105 through RFlinks 117. The first relay 105 is connected with a second relay 115 viaFSO link. The second relay 115 is connected with destinations 120through RF links 119. The abovementioned communication modalities arenon-limiting and are merely representative of a variety of communicationmodalities. The direct links between the sources and destinations areassumed to be in deep fade. Further, channel coefficients are heldconstant over an entire block of communication in accordance with ablock fading model.

Communication is operated in a half-duplex mode and to be conducted overthree phases outlined above and repeated here: selected user U_(set)→R₁,R₁→R₂, and R₂→D_(set). The received signal at R₁ 105 from the kth usercan be expressed asy _(k,r) ₁ =/√{square root over (P _(k))}h _(k,r) ₁ x _(k,r) ₁ +n _(r) ₁,  (1)where P_(k) is the transmit power of the kth user, h_(k,r) ₁ is thechannel coefficient of the U_(k)→R₁ link, x_(k,r) ₁ is the transmittedsymbol of U_(k) with

{|x_(k,r) ₁ |²}=1, and n_(r) ₁ ˜N(0, N₀₁) is an additive white Gaussiannoise (AWGN) term, where

{⋅} is the mathematical expectation. Using (1), the SNR at R₁ 105 due toU_(k) 104 can be written as

$\begin{matrix}{{\gamma\; U_{k}},{R_{1} = {\frac{P_{k}}{N_{01}}{{h_{k,r_{1}}}^{2}.}}}} & (2)\end{matrix}$

According to the generalized order user scheduling, the source with theN₁ ^(th) best γ_(U) _(k) _(,R) ₁ or equivalently, the N₁ ^(th) largest|h_(k,r) ₁ |² among the other sources 110 is selected to transmit itsmessage to R₁ 105 in the first communication phase 117. In other words,the source 110 is selected such that γ_(U) _(Sel) _(R) ₁ =N₁ ^(th)max{γ_(U) _(k) _(,R) ₁ }. Processing circuitry is configured to performthe selection process at R₁ according the selection scheme describedherein. The subcarrier intensity modulation (SIM) scheme is employed atthe relay R₁ 105, where a standard RF coherent/noncoherent modulator anddemodulator can be used for transmitting and recovering the source data.At R₁ 105, after filtering by a bandpass filter (BPF), a direct current(DC) bias is added to the filtered RF signal to ensure that the opticalsignal is non-negative. Then, the biased signal is sent to a continuouswave laser driver. The retransmitted optical signal at R₁ 105 is writtenasy _(r) ₁ ^(Opt)=√{square root over (P _(Opt))}(1+

y _(Sel,r) ₁ ),  (3)where P_(Opt) denotes the average transmitted optical power and it isrelated to the relay electrical power P_(r) by the electrical-to-opticalconversion efficiency η₁ as P_(Opt)=η₁ P_(r) ₁ , where M denotes themodulation index and γ_(Sel,r) ₁ is the RF received signal at R₁ 105from the selected source (see article by Lee, E et al, “Performanceanalysis of the asymmetric dual-hop relay transmission with mixed RF/FSOlinks” published in IEEE Transactions on Information Theory, in 2004,and incorporated herein by reference). The optical signal at R₂ 115received from R₁ 105 at the second phase of communication 118 can beexpressed asγ_(r) ₁ _(,r) ₂ =g _(r) ₁ _(,r) ₂ {√{square root over (P _(Opt))}[1+

(√{square root over (P _(Sel))}h _(Sel,r) ₁ x _(Sel,r) ₁ +n _(r) ₁ )]}+n_(r) ₂ ,  (4)where n_(r) ₂ ˜N(0, N₀₂) is an AWGN term at R₂ 115. Moreover, thechannel coefficients of the R₁→R₂ link which is given by g_(r) ₁ _(,r) ₂is modeled as g_(r) ₁ _(,r) ₂ =g_(a)g_(f), where g_(a) and g_(f) are theaverage gain and the fading gain of the FSO link, respectively, and aregiven by

$\begin{matrix}\{ \begin{matrix}{{g_{a} = {\lbrack {{erf}( \frac{\sqrt{\pi}q}{\sqrt{2}\phi\; d^{FSO}} )} \rbrack^{2} \times 10^{{- \kappa}\;{d^{FSO}/10}}}},} \\{{g_{f} \sim {{GGamma}( {\alpha,\beta} )}},}\end{matrix}  & (5)\end{matrix}$where q is the aperture radius, ϕ is the divergence angle of the beam,d^(FSO) is the distance between the FSO transmitter and receiver, κ isthe weather-dependent attenuation coefficient, and GGamma(α, β)represents a Gamma-Gamma random variable with parameters α and β (seearticle by Zhang, W et al, “Soft-switching hybrid FSO/RF links usingshort-length raptor codes: design and implementation” published in IEEEJournal on Selected Areas in Communications, in 2009, and incorporatedherein by reference). Assuming spherical wave propagation, theparameters α and β in the Gamma-Gamma distribution, which represent thefading turbulence conditions, are related to the physical parameters asfollows:

$\begin{matrix}{{\alpha = \lbrack {{\exp\{ \frac{0.49\;\vartheta^{2}}{\lbrack {1 + {0.18\;\xi^{2}} + {0.56\;\vartheta^{12/5}}} \rbrack^{7/6}} \}} - 1} \rbrack^{- 1}},} & (6) \\{{\beta = \lbrack {{\exp\{ \frac{0.51\;{\vartheta^{2}\lbrack {1 + {0.69\;\vartheta^{12/5}}} \rbrack}^{{- 5}/6}}{\lbrack {1 + {0.9\;\xi^{2}} + {0.62\;\xi^{2}\vartheta^{12/5}}} \rbrack^{5/6}} \}} - 1} \rbrack^{- 1}},} & (7)\end{matrix}$where ϑ²=0.5 C_(n) ²ζ^(7/6)(d^(FSO))^(11/6), ξ²=ζq²/d^(FSO),ζ=2π/λ^(FSO) is the wavelength, and C_(n) ² is the weather-dependentindex of refraction structure parameter (see article by He, B et al,“Bit-interleaved coded modulation for hybrid RF/FSO systems” publishedin IEEE Transactions on Communications, in 2009, and incorporated hereinby reference).

When the DC component is filtered out at R₂ 115 and anoptical-to-electrical conversion is performed, assuming M=1, thereceived signal can be expressed as follows:γ_(r) ₁ _(,r) ₂ =g _(r) ₁ _(,r) ₂ √{square root over (P_(Ele))}(√{square root over (P _(Sel))}h _(Sel,r) ₁ x _(Sel,r) ₂ +n _(r)₁ )+n _(r) ₂ ,  (8)where P_(Ele)=η₂P_(Opt)=η₁η₂P_(r) ₁ is the electrical power received atR₂ 115 and η₂ is the optical-to-electrical conversion efficiency.

From (8), the SNR at R₂ 115 can be written as

$\begin{matrix}{{{{\gamma\; R_{2}} = \frac{{\gamma\; U_{Sel}},{R_{1}\gamma\; R_{1}},R_{2}}{{\gamma\;{Sel}},{R_{1} + {\gamma\; R_{1}}},{R_{2} + 1}}},{where}}{{\gamma_{U_{Sel},R_{1}} = {\frac{P_{Sel}}{N_{01}}{h_{{Sel},r_{1}}}^{2}}},{\gamma_{R_{1},R_{2}} = {\frac{\eta_{1}\eta_{2}P_{r_{1}}}{N_{02}}{g_{r_{1},r_{2}}}^{2}}},}} & (9)\end{matrix}$and P_(r) ₁ is the transmit power at R₁ 105. The SNR in (9) can berewritten using the standard approximation γ_(R) ₂

min(γ_(U) _(Sel) _(,R) ₁ ,γ_(R) ₁ _(,R) ₂ ) (see article by Ansari, I Set al, “Impact of point errors on the performance of mixed RF/FSOdual-hop transmission systems” published in IEEE Wireless CommunicationsLetters, in 2013 and an article by Ansari, I S et al, “On theperformance of mixed RF/FSO variable gain dual-hop transmission systemswith pointing errors”, published at IEEE Vehicular TechnologyConference, in 2013, and incorporated herein by reference) as

$\begin{matrix}{{\gamma\; R_{2}} = {\frac{{\gamma\; U_{Sel}},{R_{1}\gamma\; R_{1}},R_{2}}{{\gamma\; U\;{Sel}},{R_{1} + {\gamma\; R_{1}}},{R_{2} + 1}}.}} & (10)\end{matrix}$

The signal received at D_(j) 114 from R₂ 115 in the third phase ofcommunication 119 can be written asγ_(r) ₂ _(,d) _(j) =/√{square root over (P _(r) ₂ )}h _(r) ₂ _(,j) x_(d) _(j) +n _(d) _(j) ,  (11)where P_(r) ₂ is the transmit power at R₂ 115, h_(r) ₂ _(,j) is thechannel coefficient of the R₂→D_(j) link, x_(d) _(j) is the transmittedsymbol of d_(j) with

{|x_(d) _(j) |²}=1, and n_(d) _(j) ˜N(0, N₀₃) is an AWGN term. Using(11), the SNR at D_(j) 114 can be written as

$\begin{matrix}{\gamma_{R_{2},D_{j}}\; = {\frac{P_{r_{2}}}{N_{03}}{{h_{r_{2},j}}^{2}.}}} & (12)\end{matrix}$

According to generalized order user scheduling, the destination 120 withthe N₂ ^(th) best γ_(R) ₂ _(,D) _(j) or equivalently, the N₂ ^(th)largest |h_(r) ₂ _(,j)|² among the other destinations 120 is selected toreceive its message from R₂ 115 in the third communication phase 119. Inother words, the destination is selected such that γ_(R) ₂ _(,D) _(Sel)=N₂ ^(th) max{γ_(R) ₂ _(,D) _(j) }. Processing circuitry is configuredto perform the selection process at R₂ according the selection schemedescribed herein.

The channel coefficients of the RF links 117, 119 h_(k,r) ₁ (k=1= . . .=K₁) and h_(r) ₂ _(,j) (j=1= . . . =K₂) follow the Rayleigh fading modeland, therefore, channel gains |h_(k,r) ₁ |² and |h_(r) ₂ _(,j)|² areexponentially distributed random variables with mean powers Ω_(k,r) ₁and Ω_(r) ₂ _(,j), respectively. Therefore, the probability densityfunctions (PDFs) of γ_(U) _(k) _(,R) ₁ , and γ_(R) ₂ _(,D) _(j) aregiven by

γ ⁢ ⁢ U k , R 1 ⁢ ( γ ) = λ k , r 1 ⁢ ⁢ exp ⁡ ( - λ k , r 1 ⁢ γ ) , ⁢ where ⁢ ⁢ λk , r 1 = 1 / γ _ k , r 1 ⁢ ⁢ and ⁢$\;{{{\overset{\_}{\gamma}}_{k,r_{1}} = {{\frac{P_{k}}{N_{0}}{\mathbb{E}}\{ {h_{k,r_{1}}}^{2} \}} = {\frac{P_{k}}{N_{01}}\Omega_{k,r_{1}}}}},\;{{and}\mspace{14mu}{by}}}\mspace{11mu}$γ ⁢ ⁢ R 2 , D j ⁢ ( γ ) = λ r 2 , j ⁢ ⁢ exp ⁡ ( - λ r 2 , j ⁢ γ ) , where ⁢${\lambda_{r_{2},j}\; = {{{1/{\overset{\_}{\gamma}}_{r_{2},j}}\mspace{14mu}{and}\mspace{14mu}{\overset{\_}{\gamma}}_{r_{2},j}} = {{\frac{P_{k}}{N_{03}}{\mathbb{E}}\{ {h_{r_{2},j}}^{2} \}} = {\frac{P_{r_{2}}}{N_{03}}\Omega_{r_{2},j}}}}},$respectively. Regarding the second hop, it is assumed that the FSO link118 experiences a unified Gamma-Gamma fading model including thepointing errors effect whose SNR PDF (see article by Ansari, I S et al,“Impact of point errors on the performance of mixed RF/FSO dual-hoptransmission systems” published in IEEE Wireless Communications Letters,in 2013, and incorporated herein by reference), is given by

$\begin{matrix}{{{f_{\gamma_{R_{1},R_{2}}}(\gamma)} = {\frac{\zeta^{2}}{r\;{{\gamma\Gamma}(\alpha)}{\Gamma(\beta)}}{G_{1,3}^{3,0}\lbrack {{\alpha\;{\beta( {\lambda_{r_{1},r_{2}}\gamma} )}^{\frac{1}{r}}}❘\begin{matrix}{\zeta^{2} + 1} \\{\zeta^{2},\alpha,\beta}\end{matrix}} \rbrack}}},} & (13)\end{matrix}$ζ is the ratio between the equivalent beam radius at the receiver andthe pointing error displacement standard deviation (jitter) at thereceiver (i.e. when ζ→∞, non-pointing error). r is the parameterdefining the type of detection technique (i.e. r=1 represents heterodynedetection and r=2 represents intensity modulation (IM)/direct detection(DD)). α and β are the fading parameters related to the atmosphericturbulence conditions with lower values indicating severe atmosphericturbulence conditions. Γ(.) is the Gamma function,

${\lambda_{r_{1},r_{2}} = {{1/{\overset{\_}{\gamma}}_{r_{1},r_{2}}}\mspace{14mu}{where}}}\;$$\mspace{11mu}{{\overset{\_}{\gamma}}_{r_{1},r_{2}} = {{\frac{\eta_{1}\eta_{2}P_{r_{1}}}{N_{02}}{\mathbb{E}}\{ {g_{r_{1},r_{2}}}^{2} \}} = {\frac{\eta_{1}\eta_{2}P_{r_{1}}}{N_{02}}\mu_{r_{1},r_{2}}}}}$and G(.) is the Meijer G-function (see textbook by Gradshteyn, I S andRyzhik, I M, “Tables of Integrals, Series and Products”, published byAcademic Press, in 2000, and incorporated herein by reference).

The end-to-end (e2e) SNR at the selected destination can be writtenusing the standard approximation γ_(D)

min(γ_(R) ₂ , γ_(R) ₂ _(,D) _(Sel) ) as

$\begin{matrix}{\gamma_{D} = {\frac{\gamma_{R_{2}}\gamma_{R_{2},D_{Sel}}}{\gamma_{R_{2}} + \gamma_{R_{2},D_{Sel}}}.}} & (14)\end{matrix}$

In an exemplary embodiment, as seen in FIG. 2, K₁ sources 210 are usersof wireless devices. A user U_(Sel) 204 is selected from a group ofsource users 210, as the user with the N₁ ^(th) best SNR value, totransmit its signal to a first relay 201. The signal is transmitted viaRF link on a first hop 217. Each source user (U₁ 202, U₂ 203, . . . ,U_(K) ₁ 204) is equipped with a single antenna. The first relay 201 isequipped with a single antenna 205 and a single photo-aperturetransmitter 206. A second relay 211 is equipped with a single antenna215 and a photo-aperture transmitter 216. Deploying a DF scheme, thefirst relay 201, with support from processing circuitry configured todecode a transmitted signal, receives the signal from the user U_(Sel)204 and transmits the signal to a second relay 211 on a second hop 218via FSO link. The second relay 211 receives the signal from the firstrelay 201 and, deploying a DF scheme, with support from processingcircuitry configured to decode the transmitted signal, transmits thesignal to a destination user D_(Sel) 214 via RF link on a third hop 219.The destination user D_(Sel) 214 is selected from a group of destinationusers 220, as the user with the N₂ ^(th) best SNR value, to receive itssignal from the second relay 211. Each destination user (D₁ 212, D₂ 213,. . . , D_(K) ₂ 214) is equipped with a single antenna.

System Performance Metrics (Analytical Solutions)

To evaluate system performance, the statistics of the e2e SNR providedin (14) must be determined.

To this end, the outage probability is defined as the probability thatthe SNR at a selected destination drops below a predetermined outagethreshold γ_(out), or P_(out)=Pr[γ_(d)≤γ_(out)], where Pr[.] is theprobability operation and γ_(out) is a predetermined outage threshold.The outage probability can be obtained from the cumulative distributionfunction (CDF) of the e2e SNR as P_(out)=F_(γD)(γ_(out)). This CDF canbe written in terms of CDFs of the three hops' SNRS as

$\begin{matrix}{{{F_{\gamma_{D}}(\gamma)} = {1 - \{ {( {1 - {F_{\gamma_{U_{Sel},R_{1}}}(\gamma)}} )( {1 - {F_{\gamma_{R_{1},R_{2}}}(\gamma)}} ) \times ( {1 - {F_{\gamma_{R_{2},D_{Sel}}}(\gamma)}} )} \}}},\mspace{20mu}{{where}\mspace{14mu}{F_{\gamma_{U_{Sel},R_{1}}}(\gamma)}},{F_{\gamma_{R_{1},R_{2}}}(\gamma)},{F_{\gamma_{R_{2},D_{Sel}}}(\gamma)}} & (15)\end{matrix}$are the CDFs of the first hop, second hop, and third hop SNRs,respectively.

The CDF of the first hop begins from the PDF according to generalizedorder user selection, wherein the PDF represents the N₁ ^(th) best SNRor, the source of the N₁ ^(th) best SNR as selected by the first relay.The CDF is rewritten as

$\begin{matrix}{{F_{\gamma_{U_{Sel},R_{1}}}(\gamma)} = {{K_{1}\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}}{\sum\limits_{k = 0}^{K_{1} - N_{1}}{\frac{\begin{pmatrix}{K_{1} - N_{1}} \\k\end{pmatrix}( {- 1} )^{k}}{( {k + N_{1}} )} \times {\lbrack {1 - {\exp( {{- ( {k + N_{1}} )}\lambda_{u,r_{1}}\gamma} )}} \rbrack.}}}}} & (19)\end{matrix}$where the users on the third hop have been assumed to have independentidentical distributed channels.

The CDF of the second hop is determined from the PDF of the FSO linkincorporating the Gamma-Gamma fading model and including point errors.The CDF is rewritten as

$\begin{matrix}{{{{F_{\gamma_{R_{1},R_{2}}\;}(\gamma)} = {{AG}_{{r + 1},{{3r} + 1}}^{{3r},1}\lbrack {{\frac{B}{{\overset{\_}{\gamma}}_{r_{1},r_{2}}}\gamma}❘\begin{matrix}{1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}},{where}}{{A = \frac{r^{\alpha + \beta - {2{\zeta 2}}}}{( {2\pi} )^{r - 1}{\Gamma(\alpha)}{\Gamma(\beta)}}},{B = \frac{({\alpha\beta})^{r}}{r^{2r}}},{\chi_{1} = \frac{\zeta^{2} + 1}{r}},\;{.\;.\;.}\mspace{14mu},\frac{\zeta^{2} + r}{r},}} & (20)\end{matrix}$comprises of r terms and

${\chi_{2} = \frac{\zeta^{2}}{r}},\;{.\;.\;.}\mspace{14mu},\frac{\zeta^{2} + r - 1}{r},\frac{\alpha}{r},\;{.\;.\;.}\mspace{14mu},\frac{\alpha + r - 1}{r},\frac{\beta}{r},\;{.\;.\;.}\mspace{14mu},\;\frac{\beta + r - 1}{r}$comprises of 3r terms.

Similar to the first hop, the CDF of the third hop begins from the PDFaccording to generalized order user selection, wherein the PDFrepresents the N₂ ^(th) best SNR or, the destination of the N₂ ^(th)best SNR as selected by the second relay. The CDF is rewritten as

$\begin{matrix}{{{F_{\gamma_{R_{2},D_{Sel}}}(\gamma)} = {{K_{2}\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}}{\sum\limits_{j = 0}^{K_{2} - N_{2}}{\frac{\begin{pmatrix}{K_{2} - N_{2}} \\j\end{pmatrix}( {- 1} )^{j}}{( {j + N_{2}} )} \times \lbrack {1 - {\exp( {{- ( {j + N_{2}} )}\lambda_{{r\; 2},u}\gamma} )}} \rbrack}}}},} & (22)\end{matrix}$where the users on the third hop have been assumed to have independentidentical distributed channels.

Following the substitution of the CDFs from each hop ((19), (20), (22))into (15), the full e2e CDF can be written as

$\begin{matrix}{{{{{{F_{\gamma_{D}}(\gamma)} = {{\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}K_{1}{\sum\limits_{k = 0}^{K_{1} - N_{1}}{\frac{\begin{pmatrix}{K_{1} - N_{1}} \\k\end{pmatrix}( {- 1} )^{k}}{( {k + N_{1}} )}\{ {1 - {\exp( {\tau_{1}\gamma} )} - {A \times ( {{G_{{r + 1},{{3r} + 1}}^{{3r},1}\lbrack {{\delta_{0}\gamma}❘\begin{matrix}{1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}\lbrack {1 - {\exp( {{- \tau_{1}}\gamma} )}} \rbrack} )}} \}}}} + {\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}K_{2}{\sum\limits_{j = 0}^{K_{2} - N_{2}}{\frac{\begin{pmatrix}{K_{2} - N_{2}} \\j\end{pmatrix}( {- 1} )^{j}}{( {j + N_{2}} )}\{ {1 - {\exp( {{- \tau_{2}}\gamma} )} - {A( {{G_{{r + 1},{{3r} + 1}}^{{3r},1}\lbrack {{\delta_{0}\gamma}❘\begin{matrix}{1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}\lbrack {1 - {\exp( {{- \tau_{2}}\gamma} )}} \rbrack} )}} \}}}} - {\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}K_{1}{\sum\limits_{k = 0}^{K_{1} - N_{1}}{\frac{\begin{pmatrix}{K_{1} - N_{1}} \\k\end{pmatrix}( {- 1} )^{k}}{( {k + N_{1}} )}\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}K_{2} \times}}}}}\quad}{\sum\limits_{j = 0}^{K_{2} - N_{2}}{\frac{\begin{pmatrix}{K_{2} - N_{2}} \\j\end{pmatrix}( {- 1} )^{j}}{( {j + N_{2}} )}\{ {1 - {\exp( {{- \tau_{1}}\gamma} )} - {\exp( {{- \tau_{2}}\gamma} )} + {\exp( {{- \lbrack {\tau_{1} + \tau_{2}} \rbrack}\gamma} )} - {A( {{G_{{r + 1},{{3r} + 1}}^{{3r},1}\lbrack {{\delta_{0}\gamma}❘\begin{matrix}{1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack} \times \lbrack {1 - {\exp( {{- \tau_{1}}\gamma} )} - {\exp( {{- \tau_{2}}\gamma} )} + {\exp( {{- \lbrack {\tau_{1} + \tau_{2}} \rbrack}\gamma} )}} \rbrack} )}} \}}}} + {{AG}_{{r + 1},{{3r} + 1}}^{{3r},1}\lbrack {{\delta_{0}\gamma}❘\begin{matrix}{1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}},{{{where}\mspace{14mu}\tau_{1}} = {( {k + N_{1}} )\lambda_{u,r_{1}}}},{\delta_{0} = \frac{B}{{\overset{\_}{\gamma}}_{r_{1},r_{2}}}},{{{and}\mspace{14mu}\tau_{2}} = {( {j + N_{2}} ){\lambda_{r_{2},u}.}}}} & (23)\end{matrix}$The CDF in (23) is used to determine several performance measures asclosed-form expressions.

To determine the exact average symbol error probability (ASEP), the ASEPis expressed in terms of the CDF of γ_(D) as

$\begin{matrix}{{{ASEP} = {\frac{a\sqrt{b}}{2\sqrt{\pi}}{\int_{0}^{\infty}{\frac{\exp( {{- b}\;\gamma} )}{\sqrt{\gamma}}\ {F_{\gamma_{D}}(\gamma)}d\;\gamma}}}},} & (24)\end{matrix}$where a and b are modulation-specific parameters (see article by McKay,M R et al, “Performance analysis of MIMO-MRC in double-correlatedRayleigh environments”, published in IEEE Transactions onCommunications, in 2007, incorporated herein by reference). A SIM schemeis adopted, allowing known digital modulation techniques such as phaseshift keying to be used. Therefore, the error probability computingmethod, used for RF wireless communication systems, can be used toevaluate the error probability performance in FSO systems. Uponcombination of equations, ASEP can be written as

$\begin{matrix}{{ASEP} = {\frac{a\sqrt{b}}{2\sqrt{\pi}}{\{ {{\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}K_{1}{\sum\limits_{k = 0}^{K_{1} - N_{1}}{\frac{\begin{pmatrix}{K_{1} - N_{1}} \\k\end{pmatrix}( {- 1} )^{k}}{( {k + N_{1}} )}( {\frac{\Gamma( {1/2} )}{b^{\frac{1}{2}}} - \frac{\Gamma( {1/2} )}{( {b + \tau_{1}} )^{\frac{1}{2}}} - {A\lbrack {{b^{- \frac{1}{2}}{G_{{r + 2},{{3r} + 1}}^{{3r},2}\lbrack {\frac{\delta_{0}}{b}❘\begin{matrix}{\frac{1}{2},1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}} - {( {b + \tau_{1}} )^{- \frac{1}{2}}{G_{{r + 2},{{3r} + 1}}^{{3r},2}\lbrack {\frac{\delta_{1}}{( {b + \tau_{1}} )}❘\begin{matrix}{\frac{1}{2},1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}}} \rbrack}} )}}} + {\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix} \times K_{2}{\sum\limits_{j = 0}^{K_{2} - N_{2}}{\frac{\begin{pmatrix}{K_{2} - N_{2}} \\j\end{pmatrix}( {- 1} )^{j}}{( {j + N_{2}} )}( {\frac{\Gamma( {1/2} )}{b^{\frac{1}{2}}} - \frac{\Gamma( {1/2} )}{( {b + \tau_{2}} )^{\frac{1}{2}}} - {A\lbrack {{b^{- \frac{1}{2}}{G_{{r + 2},{{3r} + 1}}^{{3r},2}\lbrack {\frac{\delta_{0}}{b}❘\begin{matrix}{\frac{1}{2},1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}} - {( {b + \tau_{2}} )^{- \frac{1}{2}}G_{{r + 2},{{3r} + 1}}^{{3r},2} \times \lbrack {\frac{\delta_{0}}{( {b + \tau_{2}} )}❘\begin{matrix}{\frac{1}{2},1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}} \rbrack}} )}}} - {\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}K_{1}{\sum\limits_{k = 0}^{K_{1} - N_{1}}{\frac{\begin{pmatrix}{K_{1} - N_{1}} \\k\end{pmatrix}}{( {k + N_{1}} )} \times ( {- 1} )^{k}\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}K_{2}{\sum\limits_{j = 0}^{K_{2} - N_{2}}{\frac{\begin{pmatrix}{K_{2} - N_{2}} \\j\end{pmatrix}( {- 1} )^{j}}{( {j + N_{2}} )}( {\frac{\Gamma( {1/2} )}{b^{\frac{1}{2}}} - \frac{\Gamma( {1/2} )}{( {b + \tau_{1}} )^{\frac{1}{2}}} - \frac{\Gamma( {1/2} )}{( {b + \tau_{2}} )^{\frac{1}{2}}} + \frac{\Gamma( {1/2} )}{( {b + \tau_{1} + \tau_{2}} )^{\frac{1}{2}}} - {A\lbrack {{b^{- \frac{1}{2}} \times {G_{{r + 2},{{3r} + 1}}^{{3r},2}\lbrack {\frac{\delta_{0}}{b}❘\begin{matrix}{\frac{1}{2},1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}} - {( {b + \tau_{1}} )^{- \frac{1}{2}}G_{{r + 2},{{3r} + 1}}^{{3r},2} \times \lbrack {\frac{\delta_{0}}{( {b + \tau_{1}} )}❘\begin{matrix}{\frac{1}{2},1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack} - {{G_{{r + 2},{{3r} + 1}}^{{3r},2}\lbrack {\frac{\delta_{0}}{( {b + \tau_{2}} )}❘\begin{matrix}{\frac{1}{2},1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack} \times ( {b \times \tau_{2}} )^{- \frac{1}{2}}} + {{G_{{r + 2},{{3r} + 1}}^{{3r},2}\lbrack {\frac{\delta_{0}}{( {b + \tau_{1} + \tau_{2}} )}❘\begin{matrix}{\frac{1}{2},1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack} \times ( {b + \tau_{1} + \tau_{2}} )^{- \frac{1}{2}}}} \rbrack}} )}}}}} + {{Ab}^{- \frac{1}{2}}{G_{{r + 2},{{3r} + 1}}^{{3r},2}\lbrack {\frac{\delta_{0}}{b}❘\begin{matrix}{\frac{1}{2},1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}}} \}.}}} & (25)\end{matrix}$

Because the coherence time of the FSO fading channel is in the order ofmilliseconds, a single fade can obliterate millions of bits atgigabits/second data rates. Therefore, the exact average (i.e., ergodic)channel capacity represents the best achievable capacity of an opticalwireless link. Using a PDF-based method, the ergodic capacity can beexpressed in terms of the PDF of γ_(D) as

$\begin{matrix}{{C = {\frac{1}{\ln(2)}{\int_{0}^{\infty}{{\ln( {1 + \gamma} )}{f_{\gamma_{D}}(\gamma)}d\;{\gamma.}}}}}\ } & (26)\end{matrix}$

Following multiple derivations and integrations described in detail inthe cited references, the exact ergodic capacity can be written as

$\begin{matrix}{C = {\frac{1}{\ln(2)}\{ {\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}K_{1}{\sum\limits_{k = 0}^{K_{1} - N_{1}}{\frac{\begin{pmatrix}{K_{1} - N_{1}} \\k\end{pmatrix}( {- 1} )^{k}}{\tau_{1}}( {{{- {\exp( \tau_{1} )}}\mspace{14mu}{E_{i}( {- \tau_{1}} )}} - {\quad{{A\{ {{\frac{1}{\delta_{0}}\lbrack {{G_{{r + 2},{{3r} + 2}}^{{{3r} + 2},1}\lbrack {\delta_{0}❘\begin{matrix}{0,1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack} - {G_{1,{0:2},{2:r},{3r}}^{0,{1:1},{2:{3r}},0} \times \lbrack {\frac{1}{\tau_{1}},{{\delta_{0}\tau_{1}}❘{\begin{matrix}1 \\ - \end{matrix}❘{\begin{matrix}{1,1} \\{1,0}\end{matrix}❘\begin{matrix}\chi_{1} \\\chi_{2}\end{matrix}}}}} \rbrack}} \rbrack}G_{1,{0:2},{2:{r + 1}},{{3r} + 1}}^{0,{1:1},{2:{3r}},1} \times  \quad{\lbrack {\frac{1}{\tau_{1}},{{\delta_{0}\tau_{1}}❘{\begin{matrix}2 \\ - \end{matrix}❘{\begin{matrix}{1,1} \\{1,0}\end{matrix}❘\begin{matrix}{1,\chi_{1}} \\{\chi_{2},0}\end{matrix}}}}} \rbrack \times ( \tau_{1} )^{2}} \}} )} + {\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}K_{2} \times {\sum\limits_{j = 0}^{K_{2} - N_{2}}{\begin{pmatrix}{K_{2} - N_{2}} \\j\end{pmatrix}( {- 1} )^{j}( \tau_{2} )^{- 1}( {{{- {\exp( \tau_{2} )}}\mspace{14mu}{E_{i}( {- \tau_{2}} )}} - {\quad{\quad{\quad{A\{ {{\frac{1}{\delta_{0}}\lbrack {G_{{r + 2},{{3r} + 2}}^{{{3r} + 2},1} \quad{\lbrack {\delta_{0}❘\begin{matrix}{0,1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack - {G_{1,{0:2},{2:r},{3r}}^{0,{1:1},{2:{3r}},0} \times  \quad\lbrack {\frac{1}{\tau_{2}},{{\delta_{0}\tau_{2}}❘{\begin{matrix}1 \\ - \end{matrix}❘{\begin{matrix}{1,1} \\{1,0}\end{matrix}❘\begin{matrix}\chi_{1} \\\chi_{2}\end{matrix}}}}} \rbrack \rbrack} + {( \tau_{2} )^{2}G_{1,{0:2},{2:{r + 1}},{{3r} + 1}}^{0,{1:1},{2:{3r}},1} \times \lbrack {\frac{1}{\tau_{2}},{{\delta_{0}\tau_{2}}❘{\begin{matrix}2 \\ - \end{matrix}❘{\begin{matrix}{1,1} \\{1,0}\end{matrix}❘\begin{matrix}{1,\chi_{1}} \\{\chi_{2},0}\end{matrix}}}}} \rbrack}} \}} )} - {\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}K_{1} \times {\sum\limits_{k = 0}^{K_{1} - N_{1}}{\frac{\begin{pmatrix}{K_{1} - N_{1}} \\k\end{pmatrix}( {- 1} )^{k}}{\tau_{1}}\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}K_{2}{\sum\limits_{j = 0}^{K_{2} - N_{2}}{( {- 1} )^{j}\frac{\begin{pmatrix}{K_{2} - N_{2}} \\j\end{pmatrix}}{\tau_{2}} \times ( {{{{- {\exp( \tau_{1} )}}\mspace{20mu}{E_{i}( {- \tau_{1}} )}} - {{\exp( \tau_{2} )}\mspace{14mu}{E_{i}( {- \tau_{2}} )}} + {{\exp( {\tau_{1} + \tau_{2}} )} \times {E_{i}( {- ( {\tau_{1} + \tau_{2}} )} )}} - {A\{ {{\frac{1}{\delta_{0}}\lbrack {{G_{{r + 2},{{3r} + 2}}^{{{3r} + 2},1}\lbrack {\delta_{0}❘\begin{matrix}{0,1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack} - {G_{1,{0:2},{2:r},{3r}}^{0,{1:1},{2:{3r}},0} \times \lbrack {\frac{1}{\tau_{1}},{{\delta_{0}\tau_{1}}❘{\begin{matrix}1 \\ - \end{matrix}❘{\begin{matrix}{1,1} \\{1,0}\end{matrix}❘\begin{matrix}\chi_{1} \\\chi_{2}\end{matrix}}}}} \rbrack} - {G_{1,{0:2},{2:r},{3r}}^{0,{1:1},{2:{3r}},0}\lbrack {\frac{1}{\tau_{2}},{{\delta_{0}\tau_{2}}❘{\begin{matrix}1 \\ - \end{matrix}❘{\begin{matrix}{1,1} \\{1,0}\end{matrix}❘\begin{matrix}\chi_{1} \\\chi_{2}\end{matrix}}}}} \rbrack} + {G_{1,{0:2},{2:r},{3r}}^{0,{1:1},{2:{3r}},0}\lbrack {\frac{1}{\lbrack {\tau_{1} + \tau_{2}} \rbrack},{{\delta_{0}\lbrack {\tau_{1} + \tau_{2}} \rbrack}❘{\begin{matrix}1 \\ - \end{matrix}❘{\begin{matrix}{1,1} \\{1,0}\end{matrix}❘\begin{matrix}\chi_{1} \\\chi_{2}\end{matrix}}}}} \rbrack}} \rbrack} + {( \tau_{1} )^{2} \times G_{1,{0:2},{2:{r + 1}},{{3r} + 1}}^{0,{1:1},{2:{3r}},1} \quad{\lbrack {\frac{1}{\tau_{1}},{{\delta_{0}\tau_{1}}❘{\begin{matrix}2 \\ - \end{matrix}❘{\begin{matrix}{1,1} \\{1,0}\end{matrix}❘\begin{matrix}{1,\chi_{1}} \\{\chi_{2},0}\end{matrix}}}}} \rbrack + {( \tau_{2} )^{2}{G_{1,{0:2},{2:{r + 1}},{{3r} + 1}}^{0,{1:1},{2:{3r}},1}\lbrack {\frac{1}{\tau_{2}},{{\delta_{0}\tau_{2}}❘{\begin{matrix}2 \\ - \end{matrix}❘{\begin{matrix}{1,1} \\{1,0}\end{matrix}❘\begin{matrix}{1,\chi_{1}} \\{\chi_{2},0}\end{matrix}}}}} \rbrack}} - {\lbrack {\tau_{1} + \tau_{2}} \rbrack^{2} \times {G_{1,{0:2},{2:r},{3r}}^{0,{1:1},{2:{3r}},0}\lbrack {\frac{1}{\lbrack {\tau_{1} + \tau_{2}} \rbrack},{{\delta_{0}\lbrack {\tau_{1} + \tau_{2}} \rbrack}❘{\begin{matrix}1 \\ - \end{matrix}❘{\begin{matrix}{1,1} \\{1,0}\end{matrix}❘\begin{matrix}\chi_{1} \\\chi_{2}\end{matrix}}}}} \rbrack}}} \}} + {\frac{1}{\delta_{0}}{G_{{r + 2},{{3r} + 2}}^{{{3r} + 2},1}\lbrack {\delta_{0}❘\begin{matrix}{0,1,\chi_{1}} \\{\chi_{2},0}\end{matrix}} \rbrack}}} \}}},} }}}}}} }}}}} }}}}}} }}} }} & (36)\end{matrix}$where E_(i)(.) is an exponential integral function, and G[Z₁, Z₂|.|.|.]is the extended generalized bivariate Meijer G-function.

System Performance Metrics (Asymptotic Solutions)

Due to the complexity of the above expressions, approximations of theseexpressions are required to appropriately evaluate the impact of changesin performance parameters and gain system insight. Detailed derivationof the approximate solutions can be found in the cited references.

The asymptotic outage probability can be written at the high SNR regimeas P_(out)

(G_(c)SNR)^(−G) ^(d) , where G_(c) and G_(d) are the coding gain anddiversity order of the system, respectively (see textbook by Simon, M Kand Alouini, M-S, “Digital Communication over Fading Channels”,published by John Wiley & Sons, Inc., in 2005, and incorporated hereinby reference). G_(c) represents a horizontal shift in the outageprobability and G_(d) refers to a change in the slope of the outageprobability vs. SNR curve.

Consider the case of identical sources' channels (λ_(1,r) ₁ =λ_(2,r) ₁ =. . . =λ_(K) ₁ _(,r) ₁ =λ_(u,r) ₁ ) and identical destinations' channels(λ_(r) ₁ _(,1)=λ_(r) ₁ _(,2)= . . . =λ_(r) ₂ _(,K) ₂ =λ_(r) ₂ _(,u)).The approximate CDF of each hop separately is determined to calculatethe approximate CDF of the e2e SNR.

Regarding the first hop link, using a Taylor series representation ofthe exponential term in the CDF to simplify and integrate, the CDF iswritten as

$\begin{matrix}{{F_{\gamma_{U_{Sel},R_{1}}}(\gamma)} \simeq {\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}{K_{1}( \lambda_{u,r_{1}} )}^{K_{1} - N_{1} + 1}{\frac{\gamma^{K_{1} - N_{1} + 1}}{( {K_{1} - N_{1} + 1} )}.}}} & (38)\end{matrix}$

Regarding the second hop link, the CDF is written as

$\begin{matrix}{{{F_{\gamma_{R_{1},R_{2}}}(\gamma)} \simeq {\Upsilon( \frac{\gamma}{{\overset{\_}{\gamma}}_{r_{1},r_{2}}} )}^{\frac{v}{r}}},} & (40)\end{matrix}$where γ is constant and is written as

$\begin{matrix}{{\Upsilon = {A{\sum\limits_{k = 1}^{6}{\frac{\prod\limits_{{j = 1},{j \neq k}}^{6}\;{{\Gamma( {b_{j} - b_{k}} )}{\Gamma( b_{k} )}}}{\prod\limits_{j = 2}^{3}\;{{\Gamma( {a_{j} - b_{k}} )}{\Gamma( {1 + b_{k}} )}}}B^{{\,^{b}k}/2}}}}},} & (43)\end{matrix}$

The third hop link, similar to the first hop link, is simplified as

$\begin{matrix}{{F_{\gamma_{R_{2},D_{Sel}}}(\gamma)} \simeq {\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}{K_{2}( \lambda_{r_{2},u} )}^{K_{2} - N_{2} + 1}{\frac{\gamma^{K_{2} - N_{2} + 1}}{( {K_{2} - N_{2} + 1} )}.}}} & (45)\end{matrix}$

To obtain the diversity order and coding gain of the system, the CDF of(16) can be simplified, at high SNR values, to beF _(γD)(γ)≅F _(γU) _(Sel) _(,R) ₁ (γ)+F _(γR) ₁ _(,R) ₂ (γ)+F _(γR) ₂_(,D) _(Sel) (γ),  (46)

Substituting values into (46), the approximate outage probability, athigh SNR values, can be written as

$\begin{matrix}{P_{out}^{\infty} = {{\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}{K_{1}( {\overset{\_}{\gamma}}_{u,r_{1}} )}^{- {({K_{1} - N_{1} + 1})}}\frac{( \gamma_{out} )^{K_{1} - N_{1} + 1}}{( {K_{1} - N_{1} + 1} )}} + ( {\frac{\Upsilon^{- \frac{r}{v}}}{y_{out}}{\overset{\_}{\gamma}}_{r_{1,}r_{2}}} )^{- \frac{v}{r}} + {\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}{K_{2}( {\overset{\_}{\gamma}}_{r_{2},u} )}^{- {({K_{2} - N_{2} + 1})}} \times {\frac{( \gamma_{out} )^{K_{2} - N_{2} + 1}}{( {K_{2} - N_{2} + 1} )}.}}}} & (47)\end{matrix}$

From (47), it is observed that the performance of the considered relaynetwork will be dominated by the worst link among the available threelinks (first RF link, FSO link, second RF link). This domination dependson the parameters of these links. Therefore, the diversity order of thetriple-hop mixed RF/FSO/RF relay network with generalized order userscheduling is equal to min(K₁−N₁+1, ν/r, K₂−N₂+1). Based on the value ofthe diversity order, one of the following three cases represents theoverall system performance.

Case 1 One hop is dominant, and the coding gain is written as

$\begin{matrix}{G_{c} = \{ {\begin{matrix}{\lbrack {\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}K_{1}\frac{( \gamma_{out} )^{K_{1} - N_{1} + 1}}{( {K_{1} - N_{1} + 1} )}} \rbrack^{- \frac{1}{K_{1} - N_{1} + 1}},} & {{G_{d} = {K_{1} - N_{1} + 1}},} \\{\frac{\Upsilon^{- \frac{r}{v}}}{y_{out}},} & {{G_{d} = \frac{v}{r}},} \\{\lbrack {\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}K_{2}\frac{( \gamma_{out} )^{K_{2} - N_{2} + 1}}{( {K_{2} - N_{2} + 1} )}} \rbrack^{- \frac{1}{K_{2} - N_{2} + 1}},} & {G_{d} = {K_{2} - N_{2} + 1}}\end{matrix}.} } & (48)\end{matrix}$

Case 2 Two hops are dominant, and the coding gain is written as

$\begin{matrix}{G_{c} = \{ \begin{matrix}{\frac{1}{2}\{ {\lbrack {\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}K_{1}\frac{( \gamma_{out} )^{K_{1} - N_{1} + 1}}{( {K_{1} - N_{1} + 1} )}} \rbrack^{- \frac{1}{K_{1} - N_{1} + 1}} +} } & {{G_{d} = {{K_{1} - N_{1} + 1} = {K_{2} - N_{2} + 1}}},} \\{ \lbrack {\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}K_{2}\frac{( \gamma_{out} )^{K_{2} - N_{2} + 1}}{( {K_{2} - N_{2} + 1} )}} \rbrack^{- \frac{1}{K_{2} - N_{2} + 1}} \},} & \; \\{{\frac{1}{2}\{ {\lbrack {\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}K_{1}\frac{( \gamma_{out} )^{K_{1} - N_{1} + 1}}{( {K_{1} - N_{1} + 1} )}} \rbrack^{- \frac{1}{K_{1} - N_{1} + 1}} + \frac{\Upsilon^{- \frac{r}{v}}}{\gamma_{out}}} \}},} & {{G_{d} = {{K_{1} - N_{1} + 1} = \frac{v}{r}}},} \\{{\frac{1}{2}\{ {\lbrack {\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}K_{2}\frac{( \gamma_{out} )^{K_{2} - N_{2} + 1}}{( {K_{2} - N_{2} + 1} )}} \rbrack^{- \frac{1}{K_{2} - N_{2} + 1}} + \frac{\Upsilon^{- \frac{r}{v}}}{\gamma_{out}}} \}},} & {{G_{d} = {{K_{2} - N_{1} + 1} = \frac{v}{r}}},}\end{matrix} } & (49)\end{matrix}$

Case 3 Three hops have the same diversity order, and so the coding gainis written as

$\begin{matrix}{G_{c} = \{ \begin{matrix}{\frac{1}{3}\{ {\lbrack {\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}K_{1}\frac{( \gamma_{out} )^{K_{1} - N_{1} + 1}}{( {K_{1} - N_{1} + 1} )}} \rbrack^{- \frac{1}{K_{1} - N_{1} + 1}} +} } & \begin{matrix}\begin{matrix}{G_{d} = {K_{1} - N_{1} +}} \\{1 = {K_{2} - N_{2} +}}\end{matrix} \\{{1 = \frac{v}{r}},}\end{matrix} \\{ {\frac{\Upsilon^{- \frac{r}{v}}}{\gamma_{out}} + \lbrack {\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}K_{2}\frac{( \gamma_{out} )^{K_{2} - N_{2} + 1}}{- ( {K_{2} - N_{2} + 1} )}} \rbrack^{- \frac{1}{K_{2} - N_{2} + 1}}} \},} & \;\end{matrix} } & (50)\end{matrix}$

System performance, dominated by the weakest link, can be described as:(1) the first hop link (i.e., K₁ and N₁), (2) the second hop link (i.e.,ζ², α, β), and (3) the third hop link (i.e., K₂ and N₂). If thediversity orders of two hops are equal and are the minimum, the codinggain of the system equals the average of the coding gains across thesetwo hops. Similarly, if the diversity orders of all three hops areequal, the coding gain of the system equals an average of the codinggains across the three hops.

The above approximate solutions are further used to determine theoptimum adaptive power allocation for the transmitting nodes in thesystem.

The distance between the first hop K₁ sources and relay R₁ is defined asd_(s,r) ₁ , the distance between the relays R₁ and R₂ is defined asd_(r) ₁ _(,r) ₂ , and the distance between relay R₂ and the third hop K₂destinations is defined as d_(r) ₂ _(,d). The distance from K₁ sourcesto K₂ destinations, therefore, is defined as D_(tot)=d_(s,r) ₁ +d_(r) ₁_(,r) ₂ +d_(r) ₂ _(,d). Under a scenario where received power decayswith distance, the average value of SNR in the hop between K₁ sourcesand relay R₁ is expressed as

${{\overset{\_}{\gamma}}_{s,r_{1}} = {P_{s,r_{1}}d_{s,r_{1}}^{- \mu}}},{{{where}\mspace{14mu} P_{{rs},_{2}}} = {\frac{P_{\mu,r_{1}}}{K_{1}} = \frac{E_{s,r_{1}}}{N_{0}}}},$μ is the path loss exponent and is equal for all hops to a value greaterthan 1, and N₀ is AWGN power (assumed equal for all hops). Similarly,average value of SNR in the second hop can be expressed as γ _(r) ₁_(,r) ₂ =P_(r) ₁ _(,r) ₂ d_(r) ₁ _(,r) ₂ ^(−μ), where

$P_{r_{1},r_{2}} = {\frac{E_{r_{1},r_{2}}}{N_{0}}.}$The average value of SNR in the third hop, between the relay R₂ anddestinations K₂, is expressed as

$P_{r_{2},d} = {\frac{P_{r_{2},d}}{K_{1}} = {\frac{E_{r_{2},d}}{K_{0}}.}}$The power constraint in this system can, therefore, be written asP_(tot)=P_(s,r) ₁ +P_(r) ₁ _(,r) ₂ +P_(r) ₂ _(,d).

The optimal power allocation, minimizing outage probability as afunction of the power constraint, is expressed as

$\begin{matrix}{{( {P_{s,r_{1}}^{*},P_{r_{1},r_{2}}^{*},P_{r_{2},d}^{*}} ) = {\arg\;{\min\limits_{({P_{s,r_{1}},P_{r_{1},r_{2}},P_{r_{2},d}})}{F_{\gamma\; D}( \gamma_{out} )}}}},} & (51)\end{matrix}$

The asymptotic expression for F_(γD)(γ_(out)) can be rewritten as

$\begin{matrix}{{F_{\gamma\; D}( \gamma_{out} )} \simeq {{\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}{K_{1}( \frac{d_{s,r_{1}}^{\mu}}{P_{s,r_{1}}} )}^{K_{1} - N_{1} + 1}\frac{( \gamma_{out} )^{K_{1} - N_{1} + 1}}{( {K_{1} - N_{1} + 1} )}} + \frac{\gamma_{out}{ABd}_{r_{1},r_{2}}^{\mu}}{P_{r_{1},r_{2}}} + {\begin{pmatrix}{K_{2} - 1} \\{N_{2} - 1}\end{pmatrix}K_{2} \times ( \frac{d_{r_{2},d}^{\mu}}{P_{r_{2},d}} )^{K_{2} - N_{2} + 1}{\frac{( \gamma_{out} )^{K_{2} - N_{2} + 1}}{( {K_{2} - N_{2} + 1} )}.}}}} & (52)\end{matrix}$

Using a Lagrangian multipliers method, differentiating, and simplifyingto solve for P_(s,r) ₁ *, P_(r) ₁ _(,r) ₂ *, and P_(r) ₂ _(,d)*, thefollowing expressions are achieved for the optimum transmission powersat each hop:

$\begin{matrix}{{P_{s,r_{1}}^{*} = \frac{d_{s,r_{1}}^{\mu}\gamma_{out}P_{tot}}{{\gamma_{out}( {d_{s,r_{1}}^{\mu} + d_{r_{2},d}^{\mu}} )} + \lbrack {{ABd}_{r_{1},r_{2}}^{\mu}\gamma_{out}} \rbrack^{\frac{1}{2}}}},} & (62) \\{{P_{r_{1},r_{2}}^{*} = \frac{\lbrack {{ABd}_{r_{1},r_{2}}^{\mu}\gamma_{out}} \rbrack^{\frac{1}{2}}P_{tot}}{{\gamma_{out}( {d_{s,r_{1}}^{\mu} + d_{r_{2},d}^{\mu}} )} + \lbrack {{ABd}_{r_{1},r_{2}}^{\mu}\gamma_{out}} \rbrack^{\frac{1}{2}}}},} & (63) \\{P_{r_{2},d}^{*} = {\frac{d_{r_{2},d}^{\mu}\gamma_{out}P_{tot}}{{\gamma_{out}( {d_{s,r_{1}}^{\mu} + d_{r_{2},d}^{\mu}} )} + \lbrack {{ABd}_{r_{2},d}^{\mu}\gamma_{out}} \rbrack^{\frac{1}{2}}}.}} & (64)\end{matrix}$

System Evaluation

The accuracy of analytical and asymptotic solutions can be validated viacomparison to Monte Carlo simulations.

FIG. 3 is a graphical representation of the impact of the order ofselected source at the first hop (N₁) and order of selected destinationat the third hop (N₂) on outage probability of the system when N₁=N₂.FIG. 3 demonstrates that under weak turbulence conditions (α=9.708 andβ=8.198), or as N₁=N₂ decreases (the quality of the selected source anddestination is improved), overall system performance is improved. Underweak turbulence, the RF links in the first and third hop drive systemperformance, and therefore improvements in SNR at those linksdramatically improves system performance. In other words, the diversityorder of the system is equal to K₁−N₁+1=K₂−N₂+1. For a fixed number ofsources and destinations (K₁=K₂), reducing N_(t)=N₂ increases thediversity order of the system and enhances system performance.

FIG. 4 is a graphical representation of the outage probability of thesystem under weak turbulence conditions when the number of sources anddestinations is varied but equal (K₁=K₂). Under weak turbulenceconditions (α=9.708 and β=8.198) or as K₁=K₂ increases (number ofavailable sources and destinations increases), overall systemperformance improves. Again, during weak turbulence, the RF links drivesystem performance and the diversity order is equal to K₁−N₁+1=K₂−N₂+1.In other words, for fixed order of selection of source and destination(N₁=N₂), an increasing K₁=K₂ increases diversity order of the system andimproves system performance.

FIG. 5 is a graphical representation of outage probability of the systemcompared with average SNR at each hop under weak turbulence conditions(α=8.650 and β=7.142) and for different values of the SNR outagethreshold γ_(out). Two specific cases were evaluated: (1) all links'average SNRs increase with increasing x-axis value and (2) one link'saverage SNR is fixed. In the case (1) where all SNRs are variable, theperformance of the system is not limited and enhances as SNR increases.In the case (2) where one link has a fixed SNR, a noise floor appears inthe results and outage probability is dominated by this worst link amongthe three. From FIG. 5, it is apparent that changes to the outagethreshold γ_(out) affect only the coding gain of the system.

FIG. 6 is a graphical representation of the outage probability of thesystem compared with order of selected source and destination (N₁=N₂)under weak turbulence conditions (α=8.038 and β=6.525) for variablevalues of average SNR per hop. As N₁=N₂ increases (as the quality of theselected source and destination decreases), the outage probability ofthe system increases and system performance degrades.

FIG. 7 is a graphical representation of the impact of the proposed powerallocation algorithm under weak turbulence conditions (α=5.662 andβ=4.059) and for variable values of outage threshold γ_(out). The dashedlines, representing power optimization, improve system performance bydecreasing outage probability of the system. Further, by increasing theoutage threshold γ_(out), system performance is degraded by reducing thecoding gain of the system and not the diversity order. In determiningpower allocation, the total distance (D_(tot)) between the sources anddestinations was assumed to be 1 and divided as D_(s,r) ₁ ₌0.3, D_(r) ₁_(,r) ₂ =0.3, and D_(r) ₂ _(, d)=0.4.

FIG. 8 is a graphical representation of the impact of pointing error (ζ)on the average symbol error probability of the system under severatmospheric turbulence conditions (α=4.341 and β=1.309). Under severeatmospheric turbulence conditions, system performance is dominated byperformance of the FSO link and its corresponding parameters (α, β, andζ²). FIG. 8 can be divided into two sets of curves where (1) diversityorder is affected by changing ζ and where (2) coding gain is affected bychanging ζ. When ζ² is the smallest parameter, G_(d) of the system isimpacted. When ζ is larger than β, changes in ζ affect the G_(c) of thesystem and G_(d) is determined by β.

FIG. 9 is a graphical representation of the impact of FSO detection type(heterodyne or intensity modulation/direct detection) on the averagesymbol error probability when compared with average SNR per hop undervariable atmospheric conditions. Due to minimal sensitivity to thermalnoise, heterodyne detection improves system performance while increasingsystem complexity.

FIG. 10 is a graphical representation of the outage probability of thesystem when the number of sources K₁ and destinations K₂ and order ofselected sources N₁ and destinations N₂ is varied. Further, theadditional impact of varied atmospheric turbulence conditions isconsidered. Under severe atmospheric turbulence conditions, increasingK₁=K₂ does not improve diversity order or coding gain of the system assystem performance is dominated by the performance of the FSO link.Under weak turbulence conditions, when the weakest links of the networkare the RF links, diversity order and coding gain are dominated by theRF link parameters (K₁, N₁, K₂, and N₂). The diversity order of thesystem is determined by the minimum value amongst the terms K₁−N₁+1 andK₂−N₂+1. Increasing only one value improves the coding gain, whileincreasing all values raises the minimum value and improves diversitygain.

FIG. 11 is a graphical representation of the ergodic capacity of thenetwork compared with average SNR per hop under weak atmosphericconditions for varying values of N₁=N₂. By increasing the quality of thesource and destination selected, or by decreasing N₁=N₂, system capacityis improved.

FIG. 12 provides a hardware description of the system for wirelessnetwork communication according to exemplary embodiments. In FIG. 12,the system for wireless network communication includes a CPU 1200 whichperforms the processes (user selection, decoding, etc.) described above.The process data and instructions may be stored in memory 1202. Theseprocesses and instructions may also be stored on a storage medium disk1204 such as a hard drive (HDD) or portable storage medium or may bestored remotely. Further, the claimed advancements are not limited bythe form of the computer-readable media on which the instructions of theinventive process are stored. For example, the instructions may bestored on CDs, DVDs, in FLASH memory, RAM, ROM, PROM, EPROM, EEPROM,hard disk or any other information processing device with which thesystem for wireless network communication communicates, such as a serveror computer.

Further, the claimed advancements may be provided as a utilityapplication, background daemon, or component of an operating system, orcombination thereof, executing in conjunction with CPU 1200 and anoperating system such as Microsoft Windows, UNIX, Solaris, LINUX, AppleMAC-OS and other systems known to those skilled in the art.

The hardware elements in order to achieve the system for wirelessnetwork communication may be realized by various circuitry elements,known to those skilled in the art. For example, CPU 1200 may be a Xenonor Core processor from Intel of America or an Opteron processor from AMDof America, or may be other processor types that would be recognized byone of ordinary skill in the art. Alternatively, the CPU 1200 may beimplemented on an FPGA, ASIC, PLD or using discrete logic circuits, asone of ordinary skill in the art would recognize. Further, CPU 1200 maybe implemented as multiple processors cooperatively working in parallelto perform the instructions of the inventive processes described above.

The system for wireless network communication in FIG. 12 also includes anetwork controller 1206, such as an Intel Ethernet PRO network interfacecard from Intel Corporation of America, for interfacing with network1230. As can be appreciated, the network 1230 can be a public network,such as the Internet, or a private network such as an LAN or WANnetwork, or any combination thereof and can also include PSTN or ISDNsub-networks. The network 1230 can also be wired, such as an Ethernetnetwork, or can be wireless such as a cellular network including EDGE,3G and 4G wireless cellular systems. The wireless network can also beWiFi, Bluetooth, or any other wireless form of communication that isknown.

The system for wireless network communication further includes a displaycontroller 1208, such as a NVIDIA GeForce GTX or Quadro graphics adaptorfrom NVIDIA Corporation of America for interfacing with display 1210,such as a Hewlett Packard HPL2445w LCD monitor. A general purpose I/Ointerface 1212 interfaces with a keyboard and/or mouse 1214 as well as atouch screen panel 1216 on or separate from display 1210. Generalpurpose I/O interface also connects to a variety of peripherals 1218including printers and scanners, such as an OfficeJet or DeskJet fromHewlett Packard.

A sound controller 1220 is also provided in the system for wirelessnetwork communication, such as Sound Blaster X-Fi Titanium fromCreative, to interface with speakers/microphone 1222 thereby providingsounds and/or music.

The general purpose storage controller 1224 connects the storage mediumdisk 1204 with communication bus 1226, which may be an ISA, EISA, VESA,PCI, or similar, for interconnecting all of the components of the systemfor wireless network communication. A description of the generalfeatures and functionality of the display 1210, keyboard and/or mouse1214, as well as the display controller 1208, storage controller 1224,network controller 1206, sound controller 1220, and general purpose I/Ointerface 1212 is omitted herein for brevity as these features areknown.

The exemplary circuit elements described in the context of the presentdisclosure may be replaced with other elements and structureddifferently than the examples provided herein. Moreover, circuitryconfigured to perform features described herein may be implemented inmultiple circuit units (e.g., chips), or the features may be combined incircuitry on a single chipset.

Moreover, the present disclosure is not limited to the specific circuitelements described herein, nor is the present disclosure limited to thespecific sizing and classification of these elements. For example, theskilled artisan will appreciate that the circuitry described herein maybe adapted based on changes on battery sizing and chemistry, or based onthe requirements of the intended back-up load to be powered.

The functions and features described herein may also be executed byvarious distributed components of a system. For example, one or moreprocessors may execute these system functions, wherein the processorsare distributed across multiple components communicating in a network.The distributed components may include one or more client and servermachines, which may share processing, in addition to various humaninterface and communication devices (e.g., display monitors, smartphones, tablets, personal digital assistants (PDAs)). The network may bea private network, such as a LAN or WAN, or may be a public network,such as the Internet. Input to the system may be received via directuser input and received remotely either in real-time or as a batchprocess. Additionally, some implementations may be performed on modulesor hardware not identical to those described. Accordingly, otherimplementations are within the scope that may be claimed.

The above-described hardware description is a non-limiting example ofcorresponding structure for performing the functionality describedherein.

It is notable that for each situation evaluated heretofore, analyticaland asymptotic expressions are in match with simulation results.

Obviously, numerous modifications and variations are possible in lightof the above teachings. It is therefore to be understood that within thescope of the appended claims, the invention may be practiced otherwisethan as specifically described herein.

Thus, the foregoing discussion discloses and describes merely exemplaryembodiments of the present invention. As will be understood by thoseskilled in the art, the present invention may be embodied in otherspecific forms without departing from the spirit or essentialcharacteristics thereof. Accordingly, the disclosure of the presentinvention is intended to be illustrative, but not limiting of the scopeof the invention, as well as other claims. The disclosure, including anyreadily discernible variants of the teachings herein, defines, in part,the scope of the foregoing claim terminology such that no inventivesubject matter is dedicated to the public.

The invention claimed is:
 1. A system for wireless networkcommunication, comprising: a first plurality of wireless devices at asource configured to transmit or receive communication viaradiofrequency; a second plurality of wireless devices at a destinationconfigured to transmit or receive communication via radiofrequency; afirst relay configured to communicate with a selected one of the firstplurality of wireless devices at the source via radiofrequency andcommunicate with a second relay via free space optical communication,the second relay being configured to communicate with the first relayvia free space optical communication and with a selected one of thesecond plurality of wireless devices at the destination viaradiofrequency; and processing circuitry configured to select a wirelessdevice at the source with the largest partially modeled sourcesignal-to-noise ratio as the selected one of the first plurality ofwireless devices, select a wireless device at the destination with thelargest partially modeled destination signal-to-noise ratio as theselected one of the second plurality of wireless devices, wherein thepartially modeled source signal-to-noise ratio is based on transmittedpower, additive white Gaussian noise, and Rayleigh fading model-basedchannel coefficients, and the partially modeled destinationsignal-to-noise ratio is based on transmitted power, additive whiteGaussian noise, and Rayleigh fading model-based channel coefficients. 2.The system for wireless network communication of claim 1, wherein theradiofrequency communication of the first plurality of wireless devicesat the source is mmWave radiofrequency communication.
 3. The system forwireless network communication of claim 1, wherein the radiofrequencycommunication of the first relay is mmWave radiofrequency communication.4. The system for wireless network communication of claim 1, wherein thefirst relay employs a decode and forward scheme.
 5. The system forwireless network communication of claim 1, wherein the second relayemploys a decode and forward scheme.
 6. The system for wireless networkcommunication of claim 1, wherein the radiofrequency communication ofthe second relay is mmWave radiofrequency communication.
 7. The systemfor wireless network communication of claim 1, wherein theradiofrequency communication of the second plurality of wireless devicesat the destination is mmWave radiofrequency communication.
 8. The systemaccording to claim 1, wherein the Rayleigh fading model-based channelcoefficients are exponentially distributed random variables.
 9. Thesystem according to claim 1, wherein the processing circuitry is furtherconfigured to calculate a partially modeled signal-to-noise ratio forthe free space optical communication, the calculated partially modeledsignal-to-noise ratio being determined in accordance with a Gamma-Gammafading model.
 10. The system according to claim 9, wherein theGamma-Gamma fading model is a unified Gamma-Gamma fading model includingpointing errors.
 11. The system according to claim 1, wherein atransmission power of the selected one of the first plurality ofwireless devices is calculated as${P_{s,r_{1}}^{*} = \frac{d_{s,r_{1}}^{\mu}\gamma_{out}P_{tot}}{{\gamma_{out}( {d_{s,r_{1}}^{\mu} + d_{r_{2},d}^{\mu}} )} + \lbrack {{ABd}_{r_{1},r_{2}}^{\mu}\gamma_{out}} \rbrack^{\frac{1}{2}}}},$where d_(s,r) ₁ ^(μ) is a distance between the source and the firstrelay, γ_(out) is a predetermined outage threshold signal-to-noiseratio, P_(tot) is a sum power constraint, d_(r) ₂ _(,d) ^(μ) is adistance between the second relay and the destination, and d_(r) ₁ _(,r)₂ ^(μ) is a distance between the first relay and the second relay. 12.The system according to claim 11, wherein an outage probability of thecommunicated signal from the selected one of the first plurality ofwireless devices to the first relay is calculated as${F_{{\gamma U}_{Sel},R_{1}}(\gamma)} = {{K_{1}\begin{pmatrix}{K_{1} - 1} \\{N_{1} - 1}\end{pmatrix}}{\sum\limits_{k = 0}^{K_{1} - N_{1}}\;{\frac{\begin{pmatrix}{K_{1} - N_{1}} \\k\end{pmatrix}( {- 1} )^{k}}{( {k + N_{1}} )} \times \lbrack {{1 - {\exp( {{- ( {k + N_{1}} )}\lambda_{u,r_{1}}\gamma} \rbrack}},} }}}$where N₁ is an order of the selected one of the first plurality ofwireless devices, K₁ is at total number of the first plurality ofwireless devices at the source, k is a number of a user, and γ is apartially modeled signal-to-noise ratio.
 13. The system according toclaim 12, wherein the transmission power of the selected one of thefirst plurality of wireless devices is determined such that the outageprobability, among the first plurality of wireless devices, of thecommunicated signal is minimized.
 14. A method of wireless networkcommunication, comprising: selecting, by processing circuitry, one of afirst plurality of wireless devices at a source, the selected one of thefirst plurality of wireless devices being a wireless device at thesource with the largest partially modeled source signal-to-noise ratio;transmitting, by the processing circuitry, a communication of theselected one of the first plurality of wireless devices to a first relayvia radiofrequency; receiving and decoding, by the processing circuitry,the transmitted radiofrequency communication from the selected one ofthe first plurality of wireless devices at the first relay; forwarding,by the processing circuitry, the decoded communication from the firstrelay to a second relay via free space optical communication; receivingand decoding, by the processing circuitry, the forwarded free spaceoptical communication from the first relay at the second relay;selecting, by the processing circuitry, one of a second plurality ofwireless devices at a destination, the selected one of the secondplurality of wireless devices being a wireless device at the destinationwith the largest partially modeled destination signal-to-noise ratio;and transmitting, by the processing circuitry, the decoded communicationfrom the second relay to the selected one of the second plurality ofwireless devices at the destination via radiofrequency, wherein thepartially modeled source signal-to-noise ratio is based on transmittedpower, additive white Gaussian noise, and Rayleigh fading model-basedchannel coefficients, and the partially modeled destinationsignal-to-noise ratio is based on transmitted power, additive whiteGaussian noise, and Rayleigh fading model-based channel coefficients.15. The method of wireless network communication of claim 14, whereinthe radiofrequency communication of the selected one of the firstplurality of wireless devices at the source is mmWave radiofrequencycommunication.
 16. The method of wireless network communication of claim14, wherein the radiofrequency communication received by the first relayis mmWave radiofrequency.
 17. The method of wireless networkcommunication of claim 14, wherein the radiofrequency communicationtransmitted by the second relay is mmWave radiofrequency.
 18. The methodof wireless network communication of claim 14, wherein theradiofrequency communication received by the selected one of the secondplurality of wireless devices at the destination is mmWaveradiofrequency.